$H^\infty$ calculus for submarkovian semigroups on weighted $L^2$ spaces

Let $(T_t)_{t \geq 0}$ be a markovian (resp. submarkovian) semigroup on some $\sigma$-finite measure space $(\Omega,\mu)$. We prove that its negative generator $A$ has a bounded $H^\infty(\Sigma_\theta)$ calculus on the weighted space $L^2(\Omega,wd\mu)$ as long as the weight $w : \Omega \to (0,\infty)$ has finite characteristic defined by $Q^A_2(w) = \sup_{t > 0} \left\| T_t(w) T_t \left(w^{-1} \right) \right\|_{L^\infty(\Omega)}$ (resp. by a variant for submarkovian semigroups). Some additional technical conditions on the semigroup have to be imposed and their validity in examples is discussed. Any angle $\theta > \frac{\pi}{2}$ is admissible in the above $H^\infty$ calculus, and for some semigroups also certain $\theta = \theta_w < \frac{\pi}{2}$ depending on the size of $Q^A_2(w)$. The norm of the $H^\infty(\Sigma_\theta)$ calculus is linear in the $Q^A_2$ characteristic for $\theta > \frac{\pi}{2}$. We also discuss negative results on angles $\theta < \frac{\pi}{2}$. Namely we show that there is a markovian semigroup on a probability space and a $Q^A_2$ weight $w$ without H\"ormander functional calculus on $L^2(\Omega,w d\mu)$.